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            <h1 style="display: none">CFD Theory</h1>
            
              <p class="note note-info">
                
                  本文最后更新于：2020年11月23日 上午
                
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            <div class="markdown-body">
              <p>Basic CFD Theory</p>
<span id="more"></span>
<h3 id="CFD-Equations"><a href="#CFD-Equations" class="headerlink" title="CFD-Equations"></a>CFD-Equations</h3><h4 id="Navier-Stokes-N-S-Equation"><a href="#Navier-Stokes-N-S-Equation" class="headerlink" title="Navier-Stokes (N-S) Equation"></a>Navier-Stokes (N-S) Equation</h4><script type="math/tex; mode=display">
\begin{align}
& \frac{\partial u_{i}}{\partial t}+ u_{j}\frac{\partial u_{i}}{\partial x_{j}}=
-\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\nu \frac{\partial^2 u_{i}}{\partial x_{j}\partial x_{j}}+g_{i}
\tag{1.1}\\
& \frac{\partial u_{i}}{\partial x_{i}}=0
\tag{1.2}

\end{align}</script><script type="math/tex; mode=display">
\begin{align}
& \frac{\partial u_{i}}{\partial t}+ u_{j}\frac{\partial u_{i}}{\partial x_{j}}=
-\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\nu \frac{\partial^2 u_{i}}{\partial x_{j}\partial x_{j}}+g_{i}
\\
& \frac{\partial u_{i}}{\partial x_{i}}=0

\end{align}</script><p>where  $u_i$ represents the air velocity, p is the pressure, $g_i$ is the gravitational acceleration.</p>
<h4 id="Energy-Equation"><a href="#Energy-Equation" class="headerlink" title="Energy Equation"></a>Energy Equation</h4><script type="math/tex; mode=display">
\begin{aligned}
&\rho \frac{\partial\left(c_{p} T\right)}{\partial t}+\rho \frac{\partial\left(c_{p} u_{j} T\right)}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\left[k \frac{\partial T}{\partial x_{j}}\right]+\Phi+S_{T}\\
&\Phi = \mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right) \frac{\partial u_{i}}{\partial x_{j}}

\end{aligned}</script><h4 id="DEM"><a href="#DEM" class="headerlink" title="DEM"></a>DEM</h4><script type="math/tex; mode=display">
\begin{aligned}
& m_{p} \frac{d u_{i}^{p}}{d t}=\sum F_{i}+\left(m_{p}-m_{f}\right) g_{i}\\
& F_{i}^{D}=\frac{\pi}{8} \rho_{f} d_{p}^{2} C_{D}\left(u_{i}^{f}-u_{i}^{p}\right)\left|u_{i}^{f}-u_{i}^{p}\right|\\
& C_{D}=\frac{24}{\operatorname{Re}_{p}}+3.6 \operatorname{Re}_{p}^{-0.313}

\end{aligned}</script><script type="math/tex; mode=display">
C_{D}=\left\{\begin{array}
{c}
\frac{24\left(1+0.15 R e^{0.687}\right)}{\operatorname{Re}} 
\text {Re} \leq 1000 \\
0.44 \mathrm{Re}>1000
\end{array}\right.</script><script type="math/tex; mode=display">
\begin{aligned}
C_{\mathrm{D}}=\left\{

\begin{array}{cc}
\frac{24}{\operatorname{Re}_{\mathrm{p}}} & \text { if } \operatorname{Re}_{\mathrm{p}} \leq 10^{3} \\
\frac{24}{\mathrm{Re}_{\mathrm{p}}}\left(1+0.15 \mathrm{Re}_{\mathrm{p}}^{0.678}\right) & \text { if } \quad \operatorname{Re}_{\mathrm{p}}>10^{3}
\end{array}\right.
\end{aligned}</script><script type="math/tex; mode=display">
m_{p} \frac{d u_{i}^{p}}{d t}=\sum F_{i}+\left(m_{p}-m_{f}\right) g_{i}\\</script><script type="math/tex; mode=display">
F_{p} = \frac{C_D Re_{p}}{24}\\</script><script type="math/tex; mode=display">
C_D = 
\left\{ 

\begin{array}
{**lr**}  
\frac{24(1+0.15Re_{p}^{0.687})}{Re_{p}} \qquad &Re_{p}\leq 10^{3} \\
0.44Re_{p}\qquad &Re_{p}<10^{3}
\end{array}

\right.</script><script type="math/tex; mode=display">
Re_{p}=\frac{\rho\left|\vec{u}-\vec{u}_{p}\right| d_{p}}{\mu_{p}}</script>
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